# Amitsur-Levitzki theorem

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A basic result in the theory of polynomial identity rings (PI-rings). A ring $R$ is a PI-ring (cf. also PI-algebra) if there is a polynomial in the free associative algebra $\mathbf{Z}\langle x_1,x_2,\ldots\rangle$ which vanishes under all substitutions of ring elements for the variables. The standard polynomial of degree $n$ is the polynomial $$S_n(x_1,\ldots,x_n) = \sum_{\sigma \in \Sigma_n} \mathrm{sign}(\sigma) \, x_{\sigma(1)}\cdots x_{\sigma(n)}$$ where $\Sigma_n$ is the symmetric group on $n$ letters. Since $S_2 = x_1 x_2 - x_2 x_1$, a ring is commutative if and only if it satisfies $S_2$ (cf. also Commutative ring). The Amitsur–Levitzki theorem says that the ring of $(n\times n)$-matrices over a commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lower degree.
The second proof, by B. Kostant [a2], depends upon the Frobenius theory of representations of the alternating group. Kostant's paper was also the first to relate the polynomial identities satisfied by matrices with traces, a theme which was later developed by C. Procesi [a4] and Yu.P. Razmyslov [a5] and influenced much research. The point is that the trace defines a non-degenerate bilinear form on $(n\times n)$-matrices.